In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable, as used in practice.
A basic example of a simple function is the floor function over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) All step functions are simple.
Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because it is easy to define integration for a simple function and also it is straightforward to approximate more general functions by sequences of simple functions.
Definition
Formally, a simple function is a finite linear combination of indicator functions of measurable sets. More precisely, let (X, Σ) be a measurable space. Let A1, ..., An ∈ Σ be a sequence of disjoint measurable sets, and let a1, ..., an be a sequence of real or complex numbers. A simple function is a function f : X → C {\displaystyle f\colon X\to \mathbb {C} } of the form
f ( x ) = ∑ k = 1 n a k 1 A k ( x ) , {\displaystyle f(x)=\sum _{k=1}^{n}a_{k}{\mathbf {1} }_{A_{k}}(x),}where 1 A {\displaystyle {\mathbf {1} }_{A}} is the indicator function of the set A.
Properties of simple functions
The sum, difference, and product of two simple functions are again simple functions, and multiplication by constant keeps a simple function simple; hence it follows that the collection of all simple functions on a given measurable space forms a commutative algebra over C {\displaystyle \mathbb {C} } .
Integration of simple functions
If a measure μ {\displaystyle \mu } is defined on the space ( X , Σ ) {\displaystyle (X,\Sigma )} , the integral of a simple function f : X → R {\displaystyle f\colon X\to \mathbb {R} } with respect to μ {\displaystyle \mu } is defined to be
∫ X f d μ = ∑ k = 1 n a k μ ( A k ) , {\displaystyle \int _{X}fd\mu =\sum _{k=1}^{n}a_{k}\mu (A_{k}),}if all summands are finite.
Relation to Lebesgue integration
The above integral of simple functions can be extended to a more general class of functions, which is how the Lebesgue integral is defined. This extension is based on the following fact.
Theorem. Any non-negative measurable function f : X → R + {\displaystyle f\colon X\to \mathbb {R} ^{+}} is the pointwise limit of a monotonic increasing sequence of non-negative simple functions.It is implied in the statement that the sigma-algebra in the co-domain R + {\displaystyle \mathbb {R} ^{+}} is the restriction of the Borel σ-algebra B ( R ) {\displaystyle {\mathfrak {B}}(\mathbb {R} )} to R + {\displaystyle \mathbb {R} ^{+}} . The proof proceeds as follows. Let f {\displaystyle f} be a non-negative measurable function defined over the measure space ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} . For each n ∈ N {\displaystyle n\in \mathbb {N} } , subdivide the co-domain of f {\displaystyle f} into 2 2 n + 1 {\displaystyle 2^{2n}+1} intervals, 2 2 n {\displaystyle 2^{2n}} of which have length 2 − n {\displaystyle 2^{-n}} . That is, for each n {\displaystyle n} , define
I n , k = [ k − 1 2 n , k 2 n ) {\displaystyle I_{n,k}=\left[{\frac {k-1}{2^{n}}},{\frac {k}{2^{n}}}\right)} for k = 1 , 2 , … , 2 2 n {\displaystyle k=1,2,\ldots ,2^{2n}} , and I n , 2 2 n + 1 = [ 2 n , ∞ ) {\displaystyle I_{n,2^{2n}+1}=[2^{n},\infty )} ,which are disjoint and cover the non-negative real line ( R + ⊆ ∪ k I n , k , ∀ n ∈ N {\displaystyle \mathbb {R} ^{+}\subseteq \cup _{k}I_{n,k},\forall n\in \mathbb {N} } ).
Now define the sets
A n , k = f − 1 ( I n , k ) {\displaystyle A_{n,k}=f^{-1}(I_{n,k})\,} for k = 1 , 2 , … , 2 2 n + 1 , {\displaystyle k=1,2,\ldots ,2^{2n}+1,}which are measurable ( A n , k ∈ Σ {\displaystyle A_{n,k}\in \Sigma } ) because f {\displaystyle f} is assumed to be measurable.
Then the increasing sequence of simple functions
f n = ∑ k = 1 2 2 n + 1 k − 1 2 n 1 A n , k {\displaystyle f_{n}=\sum _{k=1}^{2^{2n}+1}{\frac {k-1}{2^{n}}}{\mathbf {1} }_{A_{n,k}}}converges pointwise to f {\displaystyle f} as n → ∞ {\displaystyle n\to \infty } . Note that, when f {\displaystyle f} is bounded, the convergence is uniform.